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Q.
Let $a , b , c$ be the three roots of the equation $x ^3+ x ^2-333 x -1002=0$ then the value of $a^3+b^3+c^3$.
Complex Numbers and Quadratic Equations
Solution:
Let $t$ be the root of the given cubic where $t$ can take values $a, b, c$ hence $t^3+t^2-333 t-1002=0 $ or $ t^3=1002+333 t-t^2$
$\therefore \sum t ^3=\sum 1002+333 \sum t -\sum t ^2=3006+333 \sum t -\left[\left(\sum t \right)^2-2 \sum t _1 t _2\right] $
$\text { but } \sum t =-1 ; \sum t _1 t _2=-333 $
$\therefore ^3+ b ^3+ c ^3=3006-333-[1+666]=3006-333-667=3006-1000=2006$